A Jordan curve is a plane curve which is topologically equivalent to (a homeomorphic image of) the unit circle, i.e., it is simple and closed.
It is not known if every Jordan curve contains all four polygon vertices of some square, but it has been proven true
for "sufficiently smooth" curves and closed convex curves (Schnirelman
1944; Steinhaus 1999, p. 104). For every triangle 
and Jordan curve 
, 
has an inscribed triangle
similar to 
.
